Center manifolds for homoclinic solutions

Journal of Dynamics and Differential Equations

Volumn 12, Issue 3, 2000, Pages 449-510

  Sandstede, Björn ^a  

^a OHIO STATE UNIVERSITY   (United States)

Author keywords

Center manifolds; Homoclinic orbits; Shilnikov bifurcation

Indexed keywords

EID: 0007428880     PISSN: 10407294     EISSN: 15729222     Source Type: Journal    
DOI: 10.1023/A:1026412926537     Document Type: Article

References (29)

1. Belitskii, G. R. (1973). Functional equations and conjugacy of local diffeomorphisms of a finite smoothness class. Funct. Anal. Appl. 7, 268-277.
2. Bröcker, T., and Jänich, K. (1973). Introduction to Differential Topology, Springer, Berlin.
3. Brunovský, P. (1991). Personal communication.
4. Champneys, A. R. (1998). Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics. Physica D 112, 158-186.
5. Chow, S.-N., Deng, B., and Fiedler, B. (1990). Homoclinic bifurcation at resonant eigen-values. J. Dyn. Diff. Eq. 2, 177-244.
6. Chow, S.-N., and Hale, J. K. (1982). Methods of Bifurcation Theory, Springer, New York.
7. Deng, B. (1991). The bifurcations of countable connections from a twisted heteroclinic loop. SIAM J. Math. Anal. 22, 653-679.
8. Deng, B. (1991). The existence of infinitely many travelling front and back waves in the FitzHugh-Nagumo equations. SIAM J. Math. Anal. 22, 1631-1650.
9. Deng, B. (1993). On Shilnikov's homoclinic saddle-focus theorem. J. Diff. Eq. 102, 305-329.
10. Forster, O. (1981). Lectures on Riemann Surface, Springer, Heidelberg.
11. Härterich, J. (1998), Cascades of reversible homoclinic orbits to a saddle-focus equilibrium. Physica D 112, 187-200.
12. Hartman, P. (1982). Ordinary Differential Equations, Birkhäuser, Boston.
13. Henry, D. (1981). Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math., Vol. 804, Springer, New York, Berlin, Heidelberg.
14. Hirsch, M. W., Pugh, C., and Shub, M. (1977). Invariant Manifolds, Lect. Notes Math., Vol. 583, Springer, Berlin.
15. Homburg, A. J. (1996). Some Global Aspects of Homoclinic Bifurcations of Vector Fields, Memoirs of the AMS, Vol. 578, AMS, Providence, RI.
16. Homburg, H., Kokubu, H., and Krupa, M. (1994). The cusp horseshoe and its bifurcations from inclination-flip homoclinic orbits. Ergod. Theory Dyn. Syst. 14, 667-693.
17. Kato, T. (1966). Perturbation Theory for Linear Operators, Springer, Berlin.
18. Lin, X.-B. (1986). Exponential dichotomies and homoclinic orbits in functional differential equations. J. Diff. Eq. 63, 227-254.
19. Palmer, K. J. (1984). Exponential dichotomies and transversal homoclinic points. J. Diff. Eq. 55, 225-256.
20. 3. J. Diff. Eq. 126, 1-47.
21. Sandstede, B. (1993). Verzweigungstheorie homokliner Verdopplungen, Ph.D. thesis (in German), University of Stuttgart, Stuttgart.
22. Sandstede, B. (1997). Instability of localized buckling modes in a one-dimensional strut model. Phil. Trans. Roy. Soc. London A 355, 2083-2097.
23. Sandstede, B. (1998). Stability of N-fronts bifurcating from a twisted heteroclinic loop and an application to the FitzHugh-Nagumo equation. SIAM J. Math. Anal. 29, 183-207.
24. Shashkov, M. V. (1994). Bifurcations of Separatrix Contours, Ph.D. thesis (in Russian), Nizhny Novgorod State University.
25. Shashkov, M. V., and Turaev, D. Y. (1997). An existence theorem of smooth non-local center invariant manifolds for systems close to a system with a homoclinic loop. Preprint.
26. Shub, M. (1980). Global Stability of Dynamical Systems, Springer, Berlin.
27. Tresser, C. (1984). About some theorems by L. P. Shilnikov. Ann. Inst. Henri Poincare 40, 441-4161.
28. Turaev, D. Y. (1991). On Bifurcations of Dynamical Systems with Two Homoclinic Curves of the Saddle, Ph.D. thesis (in Russian), Nizhny Novgorod State University.
29. Yanagida, E. Branching of double pulse solutions from single pulse solutions in nerve axon equations. J. Diff. Eq. 66, 243-262.